Question: Estimating $\cos(1.3\pi)$ using a Taylor polynomial about $x=1.5\pi$, what is the least degree of the polynomial that assures an error smaller than $0.001$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $3$ (Choice B) B $4$ (Choice C) C $5$ (Choice D) D $6$
Solution: We will use the Lagrange error bound. Let's assume the polynomial's degree is $n$. The absolute value of the $(n+1)^{\text{th}}$ derivative of $\cos(x)$ is bounded by $1$. [Why?] The Lagrange bound for the error assures that: $|R_n(1.3\pi)|\leq \left|\dfrac{1}{(n+1)!}(1.3\pi-1.5\pi)^{n+1} \right|$ Solving $\dfrac{(0.2\pi)^{n+1}}{(n+1)!}<0.001$ using trial and error, we find that $n\geq4$. In conclusion, the least degree of the polynomial that assures our error bound is $4$.